In this appendix, we prove the regularity result stated in Proposition REF . The techniques involved are now classical and the following proof follows closely the lines of the proof of [1]}.

(i) The proof proceeds in three steps. The first step is to show the well-posedness of the function \(G(\lambda ,b,\cdot ,\cdot ,\cdot )\)  from \(\mathbb {R}\times B_{r}^{1+\alpha }\times B_{r}^{1+\alpha }\)  to \(Y^{\alpha }\)  for some \(r\)  small enough. Then, in the second step, we shall prove the existence and give the computation of the Gâteaux derivative of \(G(\lambda ,b,\cdot ,\cdot ,\cdot ).\)  Finally, in the third step, we shall prove that these Gâteaux derivatives are continuous. This will show the \(C^{1}\)  regularity of \(G(\lambda ,b,\cdot ,\cdot ,\cdot ).\)
